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Apr 5, 2022 at 3:13 vote accept Alexander Osipov
Apr 4, 2022 at 18:08 answer added Will Brian timeline score: 4
Apr 4, 2022 at 16:34 comment added Ramiro de la Vega @KPHart, note that if $\kappa=\aleph_0$ then the answer is no, regardless of the size of the continuum, since then $X\setminus \bigcup F_\alpha$ is completely metrizable and therefore it is either countable or has size continuum.
Apr 4, 2022 at 16:30 comment added KP Hart Not if the Continuum Hypothesis holds. The $k=\aleph_0$ and there is no cardinality between $\aleph_0$ and $\mathfrak{c}$.
Apr 4, 2022 at 15:18 history asked Alexander Osipov CC BY-SA 4.0